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Dynamic SPECT reconstruction from few projections: a sparsity enforced matrix factorization
approach
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2015 Inverse Problems 31 025004
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Inverse Problems
Inverse Problems 31 (2015) 025004 (26pp)
doi:10.1088/0266-5611/31/2/025004
Dynamic SPECT reconstruction from few
projections: a sparsity enforced matrix
factorization approach
Qiaoqiao Ding1, Yunlong Zan2,3, Qiu Huang2,3 and
Xiaoqun Zhang1,4
1
Department of Mathematics, MOE-LSC, Shanghai Jiao Tong University, Shanghai
200240, Peopleʼs Republic of China
2
School of Biomedical Engineering, Shanghai Jiao Tong University, Shanghai 200030,
Peopleʼs Republic of China
3
Rui Jin Hospital, School of medicine, Shanghai Jiao Tong University, Shanghai
200030, Peopleʼs Republic of China
4
Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240,
Peopleʼs Republic of China
E-mail: [email protected]
Received 11 September 2014, revised 11 November 2014
Accepted for publication 1 December 2014
Published 21 January 2015
Abstract
The reconstruction of dynamic images from few projection data is a challenging problem, especially when noise is present and when the dynamic
images are vary fast. In this paper, we propose a variational model, sparsity
enforced matrix factorization (SEMF), based on low rank matrix factorization
of unknown images and enforced sparsity constraints for representing both
coefficients and bases. The proposed model is solved via an alternating
iterative scheme for which each subproblem is convex and involves the efficient alternating direction method of multipliers (ADMM). The convergence
of the overall alternating scheme for the nonconvex problem relies upon the
Kurdyka–Łojasiewicz property, recently studied by Attouch et al (2010 Math.
Oper. Res. 35 438) and Attouch et al (2013 Math. Program. 137 91). Finally
our proof-of-concept simulation on 2D dynamic images shows the advantage
of the proposed method compared to conventional methods.
Keywords: dynamic SPECT, low rank matrix factorization, KurdykaŁojasiewicz property, sparsity, total variation
(Some figures may appear in colour only in the online journal)
0266-5611/15/025004+26$33.00 © 2015 IOP Publishing Ltd Printed in the UK
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Inverse Problems 31 (2015) 025004
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1. Introduction
Single photon emission computed tomography (SPECT) is a nuclear medicine diagnostic
technique which measures the three dimensional activity distribution of a radioactivelylabeled pharmaceutical injected into the body prior to measurement. As compared to standard
imaging techniques such as computed tomography (CT), the significance of SPECT lies in the
fact that it reveals the variation of the function rather than the structure. For example, if a
radio-pharmaceutical is rejected by healthy tissue and absorbed by unhealthy tissue, SPECT
will show the unhealthy tissue as a bright region, which helps in clinical diagnosis. Similar to
another nuclear imaging technique, positron emission tomography (PET) [1–4], SPECT has
been used in early clinical diagnosis with a higher detection sensitivity than CT or magnetic
resonance imaging (MRI). A SPECT camera works by rotating around the patient and
detecting gamma rays emitted from the radio tracer in the targeted tissue. The resulting
images on the camera are 2D projections of the original 3D activity distribution in the patient.
It is commonly assumed that in dynamic SPECT imaging the radioactivity concentration
is static during the acquisition period of each projection rotation. Many works [5–8] have
been proposed to monitor the tracer concentration functions over time under this assumption.
However, physiological processes in the body are usually dynamic, and some organs (kidney,
heart) show a significant decay of activity due to wash out, especially when the measuring
procedure takes a considerable amount of time. Hence, application of the static assumption in
the dynamic case often leads to serious motion artifacts in the reconstruction [9]. Being able
to trace activity in space and time is therefore of importance and expected to significantly
enhance diagnostic possibilities. With a significant decay of the radioisotope over the
acquisition period, the filtered back projection method (FBP) is never good in SPECT due to
the spatial attenuation of the emitted photons, and new approaches have to be developed. In
early work [10, 11], a procedure was first developed in which a dynamic image sequence was
reconstructed independently for each frame, followed by Wiener filtering across the frames
for noise reduction. Subsequently, Jin et al proposed in [12, 13] a joint reconstruction
approach for dynamic imaging, where the dynamic images of the different times are treated
collectively as a single signal through motion compensation, and determined from the
acquired data by using maximum a posteriori (MAP) estimation. Carson [14] and Formiconi
[15] put forward a method to extract the time activity curves (TACs) of regions of interest.
After that, there was a great deal of research into the extraction of the TACs of the radionuclide tracer in different tissues from a time series of reconstructed dynamic images or
directly from projections [16–18] and then estimating metabolic parameters through a compartmental model [19]. In related work, many approaches have been developed to extract the
TACs directly from SPECT projections, such as the ‘dSPECT method’ [20, 21], the ‘d2EM
method’ presented by Humphries et al [22] and factor analysis of dynamic structures (FADS)
[23]. In [9], motion of the organs has been taken into account for the dynamic CT
reconstruction.
The reconstruction of the dynamic SPECT image is an ill-posed inverse problem,
especially when very few views can be acquired. Nowadays, sparsity regularization has been
widely adopted in diverse imaging applications [24], along with rapid development of the
field of compressive sensing (CS) [25–27]. For instance, by modeling unknown images as
piecewise constant functions, total variation (TV) regularization, proposed in [28], has been
demonstrated to be effective for tomographic reconstruction from undersampled and limited
projection data [29–32]. In the dynamic setting, a low rank matrix factorization based method
is proposed in [33] for 4D cone beam CT, where sparsity, such as spatial wavelet sparsity and
temporal periodic structure for breathing cycles, are penalized for the factorization matrices.
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Inverse Problems 31 (2015) 025004
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In addition, a temporal nonlocal regularization method has also been developed in [34] for 4D
cone beam CT reconstruction from few projections data. In [35], a blurred piecewise constant
image model is proposed for reconstructing dynamic images from few projection data.
In this paper, we aim to reconstruct the dynamic radioisotope change in a SPECT image
from very few projection data (such as two projections per time interval in a pinhole camera),
under the assumption that the organs of interest are nearly static during the acquisition
procedure. Inspired by the work on dynamic image reconstruction with matrix factorization
with sparsity, such as in [33] and the concept of blind compressive sensing in [36], we
propose a matrix factorization model based on the enforcement of local and global coherence
in the temporal-spatial domain to reduce the degrees of freedom in the desired solution. More
specifically, global temporal-spatial coherence is modeled as a low rank matrix representation
U = αB⊤, where the coefficient α and the basis B carry designed physical properties. The
matrix B corresponds to the basis used in the compartmental model, from which the radioisotope concentration distribution is mixed. We also enforce the group sparsity of coefficients
by assuming that the concentration distribution is mixed from few bases for each voxel.
Meanwhile, the representation coefficients α are assumed to have small spatial and temporal
local variation according to the spatial piecewise constant model. Furthermore, the smoothness of basis TACs B is enforced through a regularization functional along the temporal
direction. Finally, the proposed variational model is composed of a data fidelity term and
several regularization elements, together with other physical constraints, such as nonnegativity.
The overall model can be alternately solved for α and B, for which each subproblem is
convex and solved by the popular l1 based first order splitting method, such as the alternating
direction method of multipliers (also known as the split Bregman method [37]). Our another
contribution is to provide a convergence analysis for the alternating scheme for the nonconvex problem. The proposed algorithm can be viewed as a proximal regularization of the
two-blocks Gauss–Seidel method. Recently, the proximal block coordinate method for
nonconvex and nonsmooth functions have been studied by many researchers, such as [38–
41]. In [38], it has been shown that if a nonconvex and nonsmooth function satisfies the
sufficient decrease condition and relative inexact optimality condition, the proximal block
descent method will have been proved to converge to a critical point without any assumption
of convexity. We will show the convergence of our scheme under this framework. Finally,
our numerical experiments on the simulated phantom have shown the feasibility of the
proposed model for reconstruction from highly undersampled data with noise, compared to
the conventional methods such as the FBP method and the least squares method. In terms of
computational complexity, the problem size has been significantly reduced due to the low
rank factorization and the subproblems are easy to implement, which allows a potential
application to real large scale reconstruction in practice.
The paper is organized as followed: section 2 briefly introduces the concept of sparse
representation with known and unknown bases. Section 3 presents the proposed model and
the numerical algorithm. The convergence is provided in section 4. Finally, section 5
demonstrates numerical results on simulated 2D dynamic images, in the settings of static
boundary, limited projections and moving boundary.
2. Sparse representation
Before presenting our model, we review the notion of sparse representation. Sparse representation or regularization has been largely used in signal and image reconstruction, machine
3
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learning and the recently developed field of compressive sensing (CS) [25–27]. The goal here
is to reconstruct a vector u ∈ n from measurements b = Au, where A ∈ m × n . This problem
is ill-posed in general and therefore has infinitely many possible solutions. A fundamental
assumption in CS is that we seek the sparsest solution:
u = arg min ∥ u ∥0
s .t .
f = Au
(1)
u
where ∥·∥0 denotes the number of nonzero elements of the vector. This idea can be
generalized to the case in which u is sparse under a given basis, for example Fourier or
wavelet bases, so that there is a sparse vector α such that u =  (α ). Problem (1) then
becomes
αˆ = arg min ∥ α ∥0
s .t .
α
f = A ( α )
(2)
and the reconstructed signal is û =  (αˆ). When the maximal number of nonzero elements in
α is known equal to k, we may consider the objective
αˆ = arg min ∥ f − A (α ) ∥22
s .t .
α
∥ α ∥0 ⩽ k .
(3)
Even when the recovery property is achieved in CS, the knowledge of the basis  is required
for the reconstruction process. In literature, dictionary leaning or sparse coding methods are
often designed to simultaneously recover both the sparse coefficients and the unknown basis
by solving the followed model
( αˆ, ˆ ) = arg min ∥ f − A (α) ∥
2
2
α, 
s .t .
∥ α ∥0 ⩽ k .
(4)
For instance, K-means singular value decomposition (K-SVD) proposed in [42] is a
popular method for learning a patch based image representation. A similar idea is also
proposed as blind compressive sensing (BCS) in [36, 43, 44]. A data driven tight frame is
proposed in [41, 45, 46] for natural image de-noising, which can greatly reduce the computational burden of the dictionary based method. All these methods are based on the
assumption that the desired signal U is a product of a sparse coefficient matrix α and a
dictionary  . In conventional sparse representation, we often have a large pool of dictionary
or analytic bases, where the degrees of freedom of the coefficients are still huge, even with the
sparsity assumption.
3. Sparsity enforced matrix factorization model
3.1. Low rank representation
The traditional tomographic problem is that f is the data observed and A is the projection
matrix and we want to reconstruct unknown image u under the condition Au = f, where the
2D/3D image u is stacked as a vector. In dynamic SPECT, we need to reconstruct a sequence
of T images u1, u 2 , ⋯uT from f1 , f2 , ⋯fT , where ft is projection data at time interval t with
different view angles. We can denote A1 , A2 , ⋯AT as T corresponding projection matrices
and
At u t = ft
t = 1, 2, ⋯ , T .
Assume that the number of pixels/voxels of each image ut is M, and the dynamic image
vector U ∈ M × T : (u1, u 2 , ⋯uT ). For ease of notation, we represent the sequence of
equations
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Inverse Problems 31 (2015) 025004
Q Ding et al
( A1 u1, A2 u 2, ⋯AT uT ) = ( f1 , f2 , ⋯fT )
(5)
as a linear transform U = f , where f is the matrix formed by the projection vectors.
Tracer kinetic modeling is widely used in biological research to make tracks for the
dynamic processes of blood flow, tissue perfusion, metabolism and receptor binding [16]. The
compartment model establishes the connection between the dynamic of the tracer with the
image and gives a mathematical description of the pathway and the dynamic behavior of the
biological tissue of interest. Between the different compartments of the model, that is the
physical space such as an organ, there exist the transportation [47] and mixture. In other
words, it assumes that the concentration distribution of the radioisotope ut(x), for each pixel/
voxel x ∈ Ω at time t can be represented as a linear combination of some basis TACs:
K
u t (x ) =
∑αk (x) Bk (t ),
k=1
where Bk(t) denotes the TAC for the kth compartment at time t, and αk (x ) denotes the mixed
coefficients. This can be reduced in matrix form as
U = αB ⊤ ,
where α ∈ M × K and B ∈ T × K with K as the number of compartments. Since, in general, K
is a small number compared to the number of time intervals T, we naturally obtain a low rank
matrix representation for U.
Note that low rank matrix factorization is a straightforward way for dimension reduction.
In particular, nonnegative matrix factorization [48–51] has been considered in many machine
learning applications to reconstruct the inline pattern or feature of the data. In general,
depending on data and applications, K can be either equal to or greater than the real rank of
the matrix. In this paper, we aim to enforce more constraints for the basis B and the coefficient
matrix α by exploring the physical meaning of these two variables in the setting of dynamic
SPECT images. Our proposed model imposes the low-rank structure of the dynamic images
by assuming that the unknown concentration distribution is a sparse linear combination of few
temporal basis functions which represent the TACs of different compartments. The dictionary
and the sparse coefficient are simultaneously recovered from the under-sampled measurements. Our model can be regarded as data driven basis learning as well as a low rank matrix
factorization approach, with further imposed constraints.
3.2. Variational model
In real problems, the observed projection data are often inevitably accompanied by noise. In
this paper, for simplicity, we consider white Gaussian noise. But we point out that the model
can be easily adapted to Poisson noise with a different data fidelity term. Suppose the data
noise is ϵ, we have
(
)
U + ϵ = ( A1 u1, A2 u 2 , ⋯AT u T ) + ϵ = f1 , f2 , ⋯fT = f ,
(6)
where U = αBT . The straightforward MAP estimator is given by solving the least squares
formulation
T
∑∥ At u t − ft ∥22
≜ ∥ αB ⊤ − f ∥2F .
t=1
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Inverse Problems 31 (2015) 025004
Q Ding et al
The generic model we propose takes the following form
∥ αB ⊤ − f ∥2F + εϕ (α ) + λψ ( αB ⊤) + σφ (B)
{ αˆ, Bˆ} = argα⩾min
0, B ⩾ 0
(7)
where ϵ, λ , σ are positive parameters. Here we impose the nonnegativity constraints for both
basis B and coefficients α so that the reconstructed image U = αB⊤ is positive. The three
terms ϕ (·) , ψ (·) and φ (·) correspond to the a priori constraints that we can impose for the
representing coefficients α, the dynamic image U = αB⊤ and the basis B. In the following, we
will explain specifically how these terms are chosen.
We set
K
ϕ (α ) =
∑TV( αk ) + κ ∥ α ∥1, ∞
(8)
k=1
to recover the desired structure in the spatial and temporal domain, where αk is the kth column
of the coefficient α, and each element of αk represents the contribution of the kth basis to the
current pixel. The ℓ1, ∞ norm is defined as
K
∥ α ∥1, ∞ =
αi , j .
∑ max
i
(9)
j=1
The total variation TV(αk ) is applied to the image form reshaped from the column vector
αk . Recall that the total variation of an image u: Ω →  is defined as
TV(u) =
∫Ω
 xu
2
2
+ yu dx ,
where  is the gradient operator. For short notation, we denote ϕ as
ϕ (α ) = ∥ α ∥1,2,1 + κ ∥ α ∥1, ∞
(10)
K
N
where ∥ v ∥1,2,1 = ∑k =1∑i,j v1,2k (i, j ) + v2,2k (i, j ) for v = α .
The first term in (8) is used to recover the piecewise constant structure since the
representing coefficients from the same structure are assumed to be the same. The second
term is designed to select as few as possible bases to represent the images ut. This type of
column sparsity was previously studied in [52] and in [50] for hyperspectral image
classification.
Considering images are piecewise constant, the nonzero gradient of the image can be
used to describe the boundary of organs. Assuming that the organs only present minor
movement or stay static, we can use the piecewise smooth structure of the edges of each
image along time. In particular, we set
(
)
(
)
ψ αB ⊤ = ∥ t  αB ⊤ ∥2F
(11)
where ∥·∥F is the Frobenius-norm of t  (αB⊤). To be more specific, denote
(
)
 αB ⊤ = U = ( u1, u 2 , ⋯u T ),
6
(12)
Inverse Problems 31 (2015) 025004
Q Ding et al
where u t = (xu t , yu t )⊤ and denote
⎛
x u 2 … x u T ⎞
⎟
y u 2 … y u T ⎠
 u
( ( )) =  (U ) =  ⎜⎝  u
t  αB ⊤
t
t
x 1
y 1
⎛ w11 w12 … w1, T ⎞
⎟
≜⎜w
≜ W.
⎝ 21 w22 … w2, T ⎠2 × T
Suppose ut is an image with resolution M = N × N , and the square of the Frobeniusnorm of W is defined as follows,
⎛ T
∑ ⎜⎜ ∑w1,2t (i, j) +
i, j ⎝ t = 1
N
∥ W ∥2F =
T
⎞
t=1
⎠
∑w2,2t (i, j) ⎟⎟.
(13)
Finally, since the decay of radioactive distribution is smooth over time, we can set
K
φ (B ) =
T
∑∑(B (t + 1, k ) − B (t, k ))2
(14)
k=1 t=1
where B (t , k ) denotes the kth basis at time t.
Taking all we have mentioned above into consideration, we can write out the model of
the problem as
K
{ αˆ, Bˆ} = arg min α∈
M × K , α ⩾ 0, B ∈ T × K , B ⩾ 0
∥ αB ⊤ − f ∥2F + ε ∑TV( αk ) + δ ∥ α ∥1, ∞
k=1
K
(
T
)
+ λ ∥ t  αB ⊤ ∥2F + σ ∑∑(B (t + 1, k ) − B (t , k ))2 ,
(15)
k=1 t=1
where ε > 0, δ > 0, λ > 0, σ > 0 and δ = εκ .
3.3. Algorithm
The optimization problem (15) is nonconvex and we solve it with a proximal Gauss–Seidel
alternating scheme on updating the coefficient α and the basis B:
(
⊤
⊤
α n + 1 = arg min α ∈ M × K , α ⩾ 0 ∥ α( B n ) − f ∥2F + εϕ (α ) + λψ α( B n )
+
L1n
2
)
(16a)
∥ α − α n ∥2F
(
)
B n + 1 = arg min B ∈ T × K , B ⩾ 0 ∥ α n + 1B ⊤ − f ∥2F + λψ α n + 1B ⊤ + σφ (B)
+
L 2n
2
(16b)
∥ B − B n ∥2F .
Here L1n ⩾ 0, L 2n ⩾ 0 are chosen parameters. Note that this scheme reduces to the usual
alternating method if the two parameters L1n = L 2n = 0 . This scheme has been studied in [38]
and [53] for functions with the so-called Kurdyka–Łojasiewicz property. Some convergence
results can be obtained, as we will discuss in detail in section 4. In [41] and [46], the
convergence of this scheme was also discussed for data driven dictionary learning models.
In the following, we will focus on solving the two convex subproblems (16a) and (16b).
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Inverse Problems 31 (2015) 025004
Q Ding et al
For convenience, we drop the subscript n for the fixed
basis matrix Bn. Denote the set characteristic function of C = {α | α ⩾ 0} as
3.3.1. Update of the coefficients α.
{
χC (α ) =
0 if α ⩾ 0,
∞ else.
(17)
The first subproblem can be rewritten as
(
)
α n + 1 = arg min ∥ αB ⊤ − f ∥2F + ε ∥ α ∥1,2,1 + δ ∥ α ∥1, ∞ + λ ∥ t  αB ⊤ ∥2F
α
L1n
∥ α − α n ∥2F .
2
The optimization model can be solved by the alternating direction method of multipliers
(ADMM) [54–56], also known as the split Bregman method [37] in imaging science. By
introducing the auxiliary variables, the model is reformulated as:
+ χC (α ) +
(
)
α n + 1 = arg min ∥ αB ⊤ − f ∥2F + ε ∥ v ∥1,2,1 + δ ∥ β ∥1, ∞ + λ ∥ t  αB ⊤ ∥2F
α
+ χC (β ) +
L1n
2
∥ α − α n ∥2F .
⎧ α = v ,
s.t. ⎨
⎩ α = β.
(18)
The constraints can be written in the linear transform:
α = (α , α )⊤ = (v , β )⊤ .
Then (18) is a convex separable optimization problem with equality constraints and can be
simplified as
arg min  (α ) +  (v , β )
α
s.t. α = (v , β )⊤ ,
(19)
L1n
2
where
 (α ) = ∥ αB −
+ λ ∥ t  (αB
+
∥α −
 (v, β ) = ε ∥ v ∥1,2,1 + δ ∥ β ∥1, ∞ + χC (β ).
For μ > 0 , the augmented Lagrangian function of the optimization problem (19) is
defined as:
μ
 (α , v , β ; p) =  (α ) +  (v , β ) + p , α − (v , β )⊤ + ∥ α − (v , β )⊤∥2F
2
where p = (p1 , p2 )⊤ is dual variable, with the same size as (v, β )⊤. Note that we can also
select a vector μ = (μ1, μ2 ) for the two constraints’ quadratic penalty, while we use the same
scalar μ here for simplicity of notation.
Thus the scheme solving (18) by ADMM is as follows:
f ∥2F
⊤
(
⊤
)
α k + 1 = arg min  α , v k , β k ; pk ,
α
(v
k + 1,
)
(
)
⊤⎞
⎛
p k + 1 = pk + μ ⎜ α k + 1 − v k + 1, β k + 1 ⎟ ,
⎝
⎠
(
α n ∥2F ,
(20a)
β k + 1 = arg min  α k + 1, v , β ; pk ,
v, β
) ∥2F
)
(20b)
(20c)
where k ∈  symbolizes the iteration steps of (20). The update for p is straightforward. In the
following, we present in detail the solution of the first two subproblems.
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Inverse Problems 31 (2015) 025004
Q Ding et al
• For subproblem (20a),
(
)
arg min  α , v k , β k ; pk = arg min  (α ) +
α
α
+
(
pk , α − v k , β k
⊤
)
⊤
μ
∥ α − v k , β k ∥2F .
2
(
)
(21)
It follows that
L1k
∥ α − α k ∥2F
2
μ
μ
+ ∥ α − v k ∥2F + p2k , α − β k + ∥ α − β k ∥2F .
2
2
(
)
α k + 1 = arg min ∥ αB ⊤ − f ∥2F + λ ∥ t  αB ⊤ ∥2F +
α
+ p1k , α − v k
The problem is quadratic and differentiable on αand the solution can be obtained by
conjugate gradient (CG) method.
• To solve (20b) for the auxiliary variables v k + 1 and β k + 1:
(v
k + 1,
)
(
β k + 1 = arg min  α k + 1, v , β ; pk
v, β
)
= arg min  (v , β ) + pk , α k + 1 − (v , β )⊤ +
v, β
μ
∥ α k + 1 − (v , β )⊤∥2F .
2
Recall that  (v, β ) = ε ∥ v ∥1,2,1 + δ ∥ β ∥1, ∞ + χC (β ) is separable on v and β. Then, v, β
can be solved separately by
(
)
(22a)
(
)
(22b)
v k + 1 = arg min  α k + 1, v , β ; pk ,
v
β k + 1 = arg min  α k + 1, v , β ; pk .
β
– For solving (22a),
(
v k + 1 = arg min  α k + 1, v , β ; pk
v
= arg min ε ∥ v ∥1,2,1
v
)
μ
+ ∥ v − v˜ k ∥2F
2
(23)
where v˜ k := μ (μ α k + 1 + p1k ). Recall that ∥ v ∥1,2,1 is defined as (10) and the solution is
obtained by soft shrinkage
1
k
⎛
ε ⎞ v˜·,t (i , j )
v·,kt+1 (i , j ) = max ⎜ ∥ v˜·,kt (i , j ) ∥2 − , 0⎟ k
μ ⎠ ∥ v˜·,t (i , j ) ∥2
⎝
⎧
⎛
ε⎞
⎪ v˜·,kt (i , j ) ⎜ ∥ v˜·,kt (i , j )∥2 − ⎟
μ⎠
⎝
ε
⎪
⎪
if ∥ v˜·,kt (i , j ) ∥2 >
k
=⎨
μ
∥ v˜·,t (i , j ) ∥2
⎪
ε
⎪0
if ∥ v˜·,kt (i , j ) ∥2 ⩽ ,
⎪
μ
⎩
where t = 1, ⋯ , K .
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Inverse Problems 31 (2015) 025004
Q Ding et al
– To solve (22b), the subproblem is simplified as
(
β k + 1 = arg min  α k + 1, v , β ; pk
β
)
K
= arg min δ ∑ max βi, j + χC (β ) +
β
i
j=1
μ
k
∥ β − β˜ ∥2F .
2
k
1
where β˜ := μ (μα k + 1 + p2k ).
From the formulation above, we can see that βis separable by column. Thus we can solve
for each column β j :
μ
∼k
β jk +1 = arg min δ max βi, j + χC β j + ∥ β j − β j ∥2F .
(24)
βj
i
2
( )
( )
The solution of this problem is given by Moreau decomposition [57–60]. The Moreau
decomposition of a convex function J in n is defined as
x = arg min J (u) +
u ∈ n
α
1
x
∥ u − x ∥22 + α arg min J * (p) + ∥ p − ∥22 ,
n
p∈
α
2α
2
where J * (p) is the conjugate function of J [58, 61]. Let J (β ) ≜ δ maxi (βi, j ) + χC (β j ), then
( )
( )
J * (p) = sup p , β j − δ max βi, j − χC β j
i
βj
= sup
βj
(
maxβi, j
i
)(
∥ max (p , 0) ∥1 − δ )
⎧ ∞ otherwise
=⎨
⎩ 0 if ∥ max (p , 0) ∥1 ⩽ δ .
(25)
From this result, we can see that J * (p) is the set characteristic function of the convex set
Cδ = {p ∈ n , ∥ max (p , 0) ∥1 ⩽ δ}. By Moreau decomposition, β k + 1 is then given by
( )
∼k
∼k
β k + 1 = β − ΠC τ β ,
(26)
∼k
∼k
where ΠCτ (β ) is the orthogonal projection of each column of β onto Cτ and τ = δμ .
The convergence of ADMM has been well studied. Numerically, we can check the
primal and dual residual for the scheme (20) as the stopping criteria [62, 63]
⎛ v k − α k ⎞
⎟,
rk = ⎜ k
⎝ β − αk ⎠
⎛ μ ⊤ vk − vk −1
⎜
dk = ⎜
k
k−1
⎝ μ β −β
(
(
)
) ⎞⎟.
⎟
⎠
3.3.2. Update of the basis B.
The update of the basis B is similar to the previous section. Let
α be the fixed nonnegative matrix α n + 1 in (16b). The second subproblem is rewritten as
B n + 1 = arg min ∥ αB ⊤ − f ∥2F + λ ∥ t αB ⊤ ∥2F + σ ∥ t B ∥2F
B
L 2n
∥ B − B n ∥2F .
2
Here we abuse the notation for the set characteristic function of the nonnegativity constraint
for B and denote ∥ t B ∥2F = ∑kK=1∑Tt=1(B (t + 1, k ) − B (t , k ))2 for simplicity of
+ χC (B) +
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Inverse Problems 31 (2015) 025004
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presentation. Analogously, we use ADMM to solve the problem by introducing an auxiliary
variable Q = B. The problem is reduced to
arg min  (B) +  (Q)
B
s.t. B = Q ,
(27)
∥ t αBT ∥2F
∥ t B ∥2F
L 2n
2
where
 (B) = ∥ αBT −
+λ
+σ
+
∥ B − B n ∥2F ,
 (Q) = χC (Q).
For conciseness of presentation, we ignore the detail and present the numerical scheme
directly as follows:
f ∥2F
B k + 1 = arg min ∥ αB ⊤ − f ∥2F + λ ∥ t αB ⊤ ∥2F + σ ∥ t B ∥2F +
B
+ P k, B − Qk +
ν
∥ B − Q k ∥2F ,
2
Q k + 1 = arg min χC (Q) +
Q
(
L 2k
∥ B − B k ∥2F
2
(28a)
ν
1
∥Q −
νB k + 1 + P k ∥2F ,
2
ν
(
)
)
P k +1 = P k + ν B k +1 − Q k +1 ,
(28b)
(28c)
where P is the dual variable.
Similarly, the subproblem (28a) is quadratic and can be solved by the conjugate gradient
∼k
1
method. For (28b), define ν (νB k + 1 + Q k ) as Q ,
Q k + 1 = arg min χC (Q) +
T
⎧
∼k
∼k
⎪
ν
∼k
Qij if Qij ⩾ 0,
∥ Q − Q ∥2F = ⎨
⎪
2
⎩0
else,
( )
∼k
≜ C Q .
(29)
The primal and dual residual of the subproblem (16b) is as follows,
rk′ = Q k − B k ,
(
)
d k′ = ν Q k − Q k − 1 .
Finally, we summarize the algorithm of SEMF model:
Algorithm 1.
Step 1. Initial value, α0, B0 , v0, β0 , Q0 , ε, δ, λ, σ , μ, ν, n = 0 . The initial dual variables p0 , P0 and the
stop condition N, δ 0
while ∥ rn ∥ > δ 0 , ∥ d n ∥ > δ 0 , ∥ rn′∥ > δ 0 , ∥ d n′∥ > δ 0 or n < N do
Step 2. Update the coefficient α
while ∥ rk ∥ > δ 0 , ∥ d k ∥ > δ 0 do
Solve (21) for α k + 1 by CG.
ε
1
v k + 1 = prox ℓ1,2,1 (v∼k , μ ), where v˜ k = μ (μ α k + 1 + p1k ).
k
k
k
∼
∼
∼
1
β k + 1 = β − ΠC δ (β ) , where β = μ (μα k + 1 + p2k +1 ).
μ
p1k +1 = p1k + μ (α k + 1 − v k + 1).
p2k +1 = p2k + μ (α k + 1 − β k + 1).
endwhile
Step 3. Update the basis B
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Inverse Problems 31 (2015) 025004
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(Continued ).
while ∥ rk′∥ > δ 0 , ∥ d k′∥ > δ 0 do
Solve (28a) for B k + 1 by CG.
∼k 1
∼k
Q k + 1 ≜ C (Q ), where Q = ν (νB k + 1 + P k ).
P k + 1 = P k + ν (B k + 1 − Q k + 1).
end while
n = n + 1.
end while
4. Convergence analysis
In this section, we will discuss the convergence of algorithm 1. In this whole section, we write
our problem (14) in a simplified form
min F (α , B) = min f (α , B) + r1 (α ) + r2 (B),
α, B
α, B
where f (α , B) is a function of α and B, r1 (α ), r2 (B) are functions of α and B respectively. In
fact, the function ∥ αBT − f ∥2F + ∥ t αBT ∥2F can be seen as f (α , B). The regularization
terms on α , B can be symbolized by r1 (α ), r2 (B), respectively. It is easy to see that f (α , B) is
a smooth function on α and B and r1 (α ) and r2 (B) are convex on α and B respectively.
Consequently, algorithm 1 can be simplified as:
Algorithm 2. Proximal alternating minimization
Initialization: α 0, B 0 , L10 > 0, L 20 > 0 .
for k = 0, 1, 2⋯ do
Lk
α k + 1 ∈ arg min α F (α, B k ) + 21 ∥ α − α k ∥22 .
B k + 1 ∈ arg min B F (α k + 1, B ) +
end for
L 2k
2
∥ B − B k ∥22 .
Note that this algorithm has been recently studied in [38, 53] under the framework of the
Kurdyka–Łojasiewicz property (see definition 2). The Kurdyka–Łojasiewicz property was
first introduced by Łojasiewicz on the real analysis function [64], and Kurdyka extended the
property to the functions on o-minimal structure [65]. The property was recently extended to
the nonsmooth subanalytic function [66].
In the following, we summarize some convergence results of algorithm 1. Denote
Z =: (α , B) and Z k =: (α k , B k ), ∀k ⩾ 0. We first present some notations and preliminaries.
Definition 1. Let f : n → ⋃ {∞} be a proper lower semicontinuous function.
(I) (Fréchet subdifferential) For each x ∈ dom(f ), the Fréchet subdifferential of f at x is the
set of vectors x* ∈ n , which satisfies
1
⎡
⎤
f (y) − f (x ) − x*, y − x ⎥ ⩾ 0
⎦
∥ x − y ∥ ⎣⎢
and can be written as ∂f̂ .
lim inf
y ≠ x, y → x
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Inverse Problems 31 (2015) 025004
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(II) (limiting-subdifferential [67]) The limiting-subdifferential of f at x ∈ dom(f ), written as
∂f , is defined as follows:
∂f (x ) :=
{ x* ∈  : ∃ x
n
n
}
→ x , f ( x n ) → f (x ), x n* ∈ ∂f̂ ( x n ) → x* .
Remark 1.
(a) From the definition we can conclude that ∂f is a close set. In fact, let (x k , x k* )k ∈  be a
sequence such that (x k , x k* ) ∈ Graph∂f for all k ∈ . If (x k , x k* ) → (x, x*) and
f (x k ) → f (x ), then (x, x*) ∈ Graph∂f .
(b) If 0 ∈ ∂f (x¯), x̄ is a critical point of f(x).
2 (The Kurdyka–Łojasiewicz (KL) property). (a) The function
f : n → ⋃ {∞} is said to have the Kurdyka–Łojasiewicz property at x* ∈ dom∂f if there
exist η ∈ (0, +∞], a neighborhood U of x* and a continuous function φ: [0, η) → +
such that:
Definition
(I)
(II)
(III)
(IV)
φ (0) = 0 ;
φ is C1 on (0, η);
for all s ∈ (0, η), φ′ (s ) > 0 ;
for all x ∈ U ⋂ [f (x*) < f < f (x*) + η], the Kurdyka–Lojasiewicz inequality holds
(
( ))dist (0, ∂f (x) ) ⩾ 1.
φ′ f (x ) − f x*
(b) If f satisfies the KL property at each point of dom(∂f ), then f is called a KL function.
For all semi-algebraic functions that are KL functions, we give the definition of a semialgebraic function.
Definition 3 (Semi-algebraic sets and functions). (a) A subset S of n is a real semi-
algebraic set if there exists a finite number of real polynomial functions gi, j , hi, j : n →  such
that
p q
{
S = ⋃⋂ x ∈ n : gi, j (x ) = 0
j i
and
}
h i , j (x ) < 0 .
(b) A function f : n → (−∞ , +∞) is called semi-algebraic if its graph,
{ (x , t ) ∈ 
}
n+1
: f (x ) = t ,
is a semi-algebraic subset of n + 1.
k
k−1
∥). Let {Z k} k ∈  be a sequence generated by
alogrithm 2 and 0 < l1 ⩽
⩽ L1 < ∞ , 0 < l2 ⩽ L 2k ⩽ L 2 < ∞. Then the sequence
k
F (Z )k ∈  is non-increasing and satisfies
Lemma 1 (square summable ∥ Z − Z
L1k
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Inverse Problems 31 (2015) 025004
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l
∥ Z k − Z k − 1 ∥2 ⩽ F Z k − 1 − F Z k , ∀k ⩾ 0,
2
(
)
( )
where l = min {l1, l2 }. Furthermore, we can deduce
∞
∑∥ Z k − Z k−1∥2
< ∞.
k=1
From lemma 1, we conclude that the objective function is decreasing for l > 0 .
This is theorem 3.1 in [38] and the property with l > 0 is called sufficient decrease.
Remark 2.
Lemma 2. f (α , B ) is LG-lipschitz continuous on a bounded set.
In
the
proposed
SEMF
model,
the
function
f (α , B) = ∥ αB⊤ − f ∥2F + λ ∥ t αB⊤ ∥2F is a smooth function. The gradient of f (α , B)
is as follows:
Proof.
f (α , B) = ( α f (α , B), B f (α , B) )
=
( ( 2
⊤
)
) )
(
+ 2λ  ⊤  t⊤ t  αB ⊤B , α ⊤ 2⊤ + 2λ   t⊤  t⊤  αB ⊤ .
f (α , B) is Lipschitz constant on any bounded set. In other words, for any bounded set S,
there exists a constant L G > 0 such that for any {(α1, B1 ), (α2, B2 )} ⊆ S ,
∥ f ( α1, B1 ) − f ( α2, B2 ) ∥ ⩽ L G ∥ ( α1, B1 ) − ( α2, B2 ) ∥ .
Then, α f (α , B) is also LG-Lipschitz continuous on the bounded set.
Lemma 3 (Relative error). {Z k} k ∈  is a sequence generated by alogrithm 2 and α f is LG-
Lipschitz continuous on a bounded set.
∥ α f ( α1, B ) − α f ( α2, B ) ∥ ⩽ L G ∥ α1 − α2 ∥ ,
∀B ∈ dom (f ) .
Then we have
(
(
dist 0, ∂F α k , B k
)) ⩽ ( L + L
G
) ∥ Z k − Z k−1∥,
where L = max {L1, L 2 }.
Proof.
See [39] theorem 3.4.
our SEMF model, F (α , B) = ∥ αB⊤ − f ∥2F + εϕ (α ) + λψ (αB⊤)
+ σφ (B) + χC (α ) + χC (B) satisfies the KL property.
Lemma
4. In
For ∥ αBT − f ∥2F , ∥ t αBT ∥2F and ∥ t B ∥2F are real polynomial functions,
∥ αB − f ∥2F , ψ (αB⊤) and φ (B) are semi-algebraic functions [39].
Proof.
T
The
nonnegative
orthant
+M ×K = {α ∈ M × K : αi, j ⩾ 0, ∀i, j},
T+×K = {B ∈ M × K : Bi, j ⩾ 0, ∀i, j} are semi-algebraic sets. For indicator functions of semialgebraic sets that are semi-algebraic, χC (α ) and χC (B) are semi-algebraic functions.
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Inverse Problems 31 (2015) 025004
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For the function ϕ (α ) = ε ∥ α ∥1,2,1 + λδ ∥ α ∥1, ∞, we denote v = α , then,
∥ v ∥1,2,1 = ∑kK=1∑iN,j
v1,2k (i, j ) + v2,2k (i, j ) . The graph of
( v (i , j ) + v (i , j ) )
= { ( v (i , j ), v (i , j ), t ) ∈  :
2
1,k
Graph
1, k
=
{(v
1, k (i ,
v1,2k (i, j ) + v2,2k (i, j ) is
2
2,k
}
v1,2k (i , j ) + v2,2k (i , j ) = t
3
2, k
j ), v2, k (i , j ), t ) ∈  :
3
v1,2k (i ,
j) +
v2,2k (i ,
j) =
t 2,
}
t⩾0 .
It is easy to know that Graph(
j) +
j ) ) is a semi-algebraic set. For finite sums
and products of semi-algebraic functions that are semi-algebraic functions, we know ∥ v ∥1,2,1
is a semi-algebraic function. Then, we conclude that ∥ α ∥1,2,1 is a semi-algebraic function for
the composition of semi-algebraic functions or mappings are semi-algebraic. For
∥ α ∥1, ∞ = ∑Kj=1 maxi αi, j , the graph of maxi αi, j is
v1,2k (i,
(
Graph max αi, j
i
) {
=
v2,2k (i,
} = ⋃ { (α (·, j), t): α
(α (·, j ), t ) : max αi, j = t
i
i, j
= t , αk , j ⩽ t ,
i
∀k ≠ i},
which is a semi-algebraic set. ∥ α ∥1, ∞ is a semi-algebraic function since it is finite sums of
semi-algebraic functions.
From all of the above, F (α , B) is the finite sum of the semi-algebraic functions and
satisfies the KL property.
Theorem 4 (Lemma 2.6 [40]). {Z k} k ∈  is a sequence generated by proximal alternating
minimization and 0 < l1 ⩽ L1k ⩽ L1 < ∞ , 0 < l2 ⩽ L 2k ⩽ L 2 < ∞. Moveover, the following hold:
(i) f is LG-Lipschitz continuous on a bounded set;
(ii) F satisfies the KL inequality at Z̄ ;
(iii) Z0 is sufficiently close to Z̄ , and F (Z k ) > F¯ for k ⩾ 0 .
Then there exists  (Z¯ , ρ) ∈ U ⋂ dom(∂F ), such that {Z k} ⊂  and Zk converges to
Z * ∈  and
∞
∑ ∥ Z k +1 − Z k ∥ < ∞,
k=0
( )
F Z k → F ( Z¯ ), as
k → ∞,
where U is defined in definition 2 relative to F and Z̄ .
Theorem 5 (convergence to a critical point). Let {Z k} k ∈  be a sequence generated by
algorithm 2 and 0 < l1 ⩽ L1k ⩽ L1 < ∞ , 0 < l2 ⩽ L 2k ⩽ L 2 < ∞. Z0 is sufficiently close to
Z̄, and Fk > F̄ for k ⩾ 0 . Let ρ > 0 such that  (Z¯ , ρ) ⊂ U , then {Z k} ⊂  and Zk
converges to Z * ∈  . Furthermore, if Zk is bounded, then Zk converges to a critical point Z*
of F(Z).
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Inverse Problems 31 (2015) 025004
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From lemma 4, our SEMF model satisfies the KL property and from lemma 2, f is
LG-Lipschitz continuous on a bounded set. Then by theorem 4, we can conclude that
 (Z¯ , ρ) ∈ U ⋂ dom(∂F ), such that {Z k} ⊂  and Zk converges to Z * ∈  .
Proof.
From above, Z * = (α*, B*) is a limit point of {Z k} k ∈  = {(α k , B k )}k ∈ . Then, there
exists a subsequence {(α k q, B k q )}q ∈  such that (α k q, B k q ) → (α*, B*). For our problem, we
can easily obtain that lim q →∞ F (α k q, B k q ) = F (α*, B*). Furthermore, by the property of
relative error in lemma 3, we know that ∂F (α k q, B k q ) → (0, 0). For ∂F is a close convex set
and by the definition limiting-subdifferential, we can get (0, 0) ∈ ∂F (α*, B*). Consequently,
Z* is a critical point of F(Z).
Define the difference measure of two sets  , 
⎛
⎞
diff( ,  ) = max ⎜ sup inf ∥ α − B ∥ , sup inf ∥ α − B ∥⎟ .
⎝ α∈ B∈
⎠
B∈α∈
The set of feasible points of F (α , B),  , is a closed and block multiconvex subset of n ,
which can be decomposed into two blocks. We define
1 (B) ≜ {α : (α , B) ∈ },
2 (α ) ≜ {B : (α , B) ∈ }.
(Nash point). A point x̄ is called the Nash point of a proper lower
semicontinuous function f (x1, x 2 , ⋯x s ): n → , if for i = 1, ⋯ , s and feasible point set  ,
Definition 6
f ( x¯1, ⋯ , x¯i − 1, x¯i , x¯i + 1 ⋯x¯s ) ⩽ f ( x¯1, ⋯ , x¯i − 1, xi , x¯i + 1 ⋯x¯s )
∀xi ∈ ¯ i
or equivalently,
∂ x if ( x¯), xi − x¯i ⩾ 0
∀xi ∈ ¯ i,
where ¯ i = {xi ∈ ni : (x¯1, ⋯ , x¯i − 1, xi , x¯i + 1 ⋯x¯s ) ∈ } and ni is the dimension of the ith
block.
Theorem 7. If Z k , Z ∈  and Z k → Z implies
( ( )
)
( ( )
)
lim diff 1 B k , 1 (B) = 0 lim diff 2 α k , 2 (α ) = 0.
k →∞
k ′→∞
Then if ∥ Z k + 1 − Z k ∥ → 0 holds, any limit point of {Z k} k ∈  is a Nash point.
Proof.
See theorem 2.3 in [40].
Since each subproblem of algorithm 1 is convex, we can also conclude that
Theorem 8 (limit point is a Nash point). Let {Z k} k ∈  be a sequence generated by
algorithm 2. If Zk is bounded, then any limit point of Zk is a Nash point.
j
Let Z* be a limit point of Zk, then there exists a subsequence Z k → Z *. For
j
convenience, we write Z k′ as Z k . We can get that
Proof.
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Inverse Problems 31 (2015) 025004
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Figure 1. The TAC of blood, liver and myocardium.
( ( ) ) ( )
lim diff (  ( α ) ,  (α ) ) = diff ( B , B*) = 0.
lim diff 1 B k ′ , 1 (B) = diff α k ′, α* = 0,
k ′→∞
k ′→∞
2
k′
k′
2
k
k−1 2
From lemma 1, ∑∞
∥ < ∞, then we have ∥ Z k − Z k − 1 ∥ → 0. Referring to
k =1 ∥ Z − Z
theorem 7, we obtain the conclusion that any limit point of Zk is a Nash point.
5. Numerical results
5.1. 2D dynamic images with static boundary
Now, we present numerical results to verify our model and algorithm. The algorithm is tested
on numerical phantoms for a proof of concept study. We simulate 90 image frames of size
64 × 64 and two projections per frame. Three time activity curves (TAC) for blood, liver and
myocardium, previously used in [18] (see figure 1), are used to simulate the dynamic images.
The first simulated dynamic phantom is composed of two ellipses. In the temporal direction,
the positions of the two ellipses are stationary while the intensity in 90 frames within the
region of each ellipse is generated according to the TAC of blood or liver. Then the projections are generated by Radon transforms, sequentially performed for each frame.
We compare our method with the filtered back projection (FBP) method, the solutions by
solving two least square models argminU ∥ U − f ∥2F , arg min α ∥ αB⊤ − f ∥2F with a given
fixed B-spline basis B.
As for the initial values of α and B, we use 16 uniform B-spline bases to solve
argmin α ∥ αB⊤ − f ∥2F , where the knots of the B-spline basess are uniformly distributed over
0–90 s. Then, the solution α obtained by solving argmin α ∥ αB⊤ − f ∥2F and 16 uniform Bspline bases, B, are used as the initialization of α and B to solve our SEMF model.
In the first test, projections at two orthogonal angles are simulated for every frame to
mimic two-head camera data collection. The projection angles increase sequentially by 1°
along the temporal direction. For example, at frame 1, projections are simulated at angle 1°
and 91°, and at frame 2, angle 2° and 92°, etc. Finally, 10% white Gaussian noise is added to
the projection data. Reconstruction results using different methods are shown in figure 2.
Since the number of projections is very limited for each frame, the traditional FBP and least
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Inverse Problems 31 (2015) 025004
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Figure 2. From top to bottom: true image frames; reconstructed by FBP; a solution
solving argminU ∥ U − f ∥2F ; a solution using a fixed basis B, solving
argmin α ∥ αB⊤ − f ∥2F ; reconstructed by our method.
Figure 3. The TACs of blood and liver. (a) is the TAC of blood and (b) is the TAC of
liver. The solid lines are the true TACs and the dash lines are the reconstructed TACs.
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Inverse Problems 31 (2015) 025004
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Figure 4. The TACs used in the rats’ liver image.
Figure 5. From top to bottom: true image frames; reconstruction by FBP; a solution
solving argminU ∥ U − f ∥2F ; a solution using a fixed basis B, solving
argmin α ∥ αB⊤ − f ∥2F ; reconstructed by our method.
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Inverse Problems 31 (2015) 025004
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Figure 6. The true TAC and the TAC we obtained from the SEMF model. In the four
figures, the solid lines are the true TACs and the dash lines are the reconstructed TACs.
Figure 7. The rates of ∥ Urec − Utrue ∥22 and ∥ Utrue ∥22 . The left is the rate of the first
numerical experiment and the right is the rate of the second.
squares methods cannot reconstruct the images satisfactorily, while the proposed method is
capable of reconstructing the images effectively.
Figure 3 illustrates the comparison of the TACs of blood and liver. The solid lines are the
true TACs and the dash lines are the ones extracted from the reconstruction images by our
method. Due to the weak signal and fast change of radioisotope at the beginning frames in the
liver TAC, the reconstructed one does not fit closely to the true one.
The second numerical experiment is performed on a synthetic image simulating a ratʼs
abdomen, where the bright region represents the liver of a rat. We use the TAC in figure 4 to
simulate the dynamic images. The same procedure is applied to generate projection data.
Figure 5 compares the frames reconstructed by different methods. Clearly, the traditional FBP
method and least squares method cannot reconstruct the dynamic images with very few
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Q Ding et al
Figure 8. The objective function of the model. The left is the first numerical experiment
and the right is the second one.
Figure 9. The result of two-dimensional images with one projection angle. The images
are reconstructed using SEMF and every frame has one projection uniformly sampled
from [0, 180]. The first row is the result of ellipse and the second row is the result of the
ratʼs liver.
Figure 10. The result of two-dimensional images with the projection angular space
[30, 150]. The images are reconstructed by our SEMF model and every image has two
projection angles uniformly sampled from [30, 150]. The first row is the result of the
ellipse and the second row is the result of the ratʼs liver.
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Q Ding et al
Figure 11. The area variation of the two ellipses by time.
projections; however, the proposed method reconstructs the images quite accurately. Similar
to the ellipse phantom, figure 6 illustrates the comparison of the true TACs and those
reconstructed by the SEMF method. We can see that two of them are quite accurate while the
other two present relatively larger errors.
2
Figure 7 demonstrates the relative error ∥ U (:, t )rec − U (:, t2)true ∥2 for the tth frame, where Urec is
∥ U (:, t )true ∥2
the reconstructed frame by the proposed method and Utrue is the ground truth image. We can
see that the relative error is larger for the first frames. This is partly due to the fact that the
intensity of the image is in a much smaller scale and the decay of activity is more significant
for these frames. Figure 8 demonstrates the evolution of the total energy of the model. It has
been proved that the energy is decreasing in lemma 1 and numerically we can observe the
global decrease with small numerical errors.
In the following, we test the proposed method on two more challenging settings: one is
with only one projection per frame and the other is with limited projection angles which does
not cover the whole range of [0, 180]. For the first test, the total number of projections is 90
compared to 180 in the previous tests for 90 frames, while the angles are uniformly sampled
from the angular range [0, 180]; and for the second test there are two projections per frame
with angular distance 60° and the angles are uniformly sampled from the range [30, 150]. We
add a further 1% noise to the projection data of the ellipse and 10% noise to the one of the
ratʼs liver, respectively. Figures 9 and 10 show the reconstruction results on the two test
images by the SEMF method, from one-projection-per-frame and partial angles settings
respectively. We can see that with even less data, our method can recover most of the
information, especially for those frames at the stable stage. In figure 10, the images suffer
more stripe effects due to the limited angles, while the default is partially overcome by the
symmetricity for the ellipse image.
5.2. 2D dynamic images with moving boundaries
In practice, the boundary of the observed organ is moving with time due to the respirator
motion. In this section, we verify the feasibility of the proposed model for such a scenario.
Similar to the previous experiment, we simulate 90 frames of images composed of two
ellipses with the intensity changing according to the TACs. Furthermore, we simulate a
periodic motion for the boundary of the ellipses. In particular, the semi-major and semi-minor
axes of the ellipses change cyclically. The area variation of the two ellipses is shown in
figure 11. The projection data of each image in the two angles are simulated and 10% white
Gaussian noise is also added.
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Inverse Problems 31 (2015) 025004
Q Ding et al
Figure 12. The simulated image sequence of two-dimensional images with dynamic
boundaries. The first row is the true image, the second row is the images reconstructed
by FBP and the third row is the solution of argminU ∥ U − f ∥2F . The forth row is
fixed basis B, solved argmin α ∥ αB⊤ − f ∥2F and the last row is the images
reconstructed by our model.
The comparison of the results of the different reconstruction methods are shown in
figure 12. We can see that our model can reconstruct the images even with motion present,
while the other methods fail. For this test case, similar curves can be obtained for the the
TACs, relative error and the decreasing energy as shown in figures 3, 7 and 8.
6. Conclusion
In this paper, we presented a novel reconstruction model for dynamic SPECT that combines
low rank decomposition and temporal and spatial regularization. The model greatly reduces
the degree of freedom of the solution, which allows the reconstruction of the dynamic images
with very few and incomplete projections. We also provide a convergence analysis for the
alternating scheme to solve the proposed nonconvex model. Our reconstruction results on a
2D phantom with three examples, two with static boundaries and one with moving boundaries, indicate that our algorithm outperforms the conventional FBP-type reconstruction
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Inverse Problems 31 (2015) 025004
Q Ding et al
algorithm and least squares method. This study has verified the feasibility of such a regularization model for dynamic image reconstruction from few projections, and allows for the
future implementation and adaptation to real 3D reconstruction with Poisson noise.
Acknowledgments
We are grateful to the anonymous reviewers for their valuable comments and advice. Q Ding
and X Zhang were partially supported by the NSFC (grant nos. NSFC11101277 and
NSFC91330102) and the 973 program (no. 2015CB856004). Y Zan and Q Huang were
partially supported by the NSFC (grant nos. NSFC81201114 and NSFC91330102) and the
Innovation Program of Shanghai Municipal Education Commission (grant no. 13ZZ017).
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